Assessment of Solutions to Vibration Problems Involving Piecewise Constant Exertions

Numerical Solutions ◽

Numerical Calculations ◽

Piecewise Constant ◽

Truncation Errors ◽

Vibration Problems ◽

Linear And Nonlinear ◽

Abstract Direct analytical and numerical solutions are constructed for linear and nonlinear vibration problems involving piecewise constant exertions. Existence and uniqueness of the solutions and the truncation errors of the numerical calculations are also analysed. With the employment of a piecewise constant argument, vibration systems with piecewise constant exertions are connected with the corresponding systems with continuous exertions.

Existence and Uniqueness of Periodic Solutions for a Second-Order Nonlinear Differential Equation with Piecewise Constant Argument

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/2009/950797 ◽

2009 ◽

Vol 2009 ◽

pp. 1-14 ◽

Cited By ~ 4

Author(s):

Gen-qiang Wang ◽

Sui Sun Cheng

Keyword(s):

Differential Equation ◽

Periodic Solution ◽

Periodic Solutions ◽

Nonlinear Differential Equation ◽

Second Order ◽

Piecewise Constant ◽

Order Nonlinear Differential Equation ◽

Based on a continuation theorem of Mawhin, a unique periodic solution is found for a second-order nonlinear differential equation with piecewise constant argument.

Emerging Applications of Differential Equations and Game Theory - Advances in Computer and Electrical Engineering ◽

Stability Analysis of a Nonlinear Epidemic Model With Generalized Piecewise Constant Argument

10.4018/978-1-7998-0134-4.ch009 ◽

2020 ◽

pp. 182-208

Author(s):

Duygu Aruğaslan-Çinçin ◽

Nur Cengiz

Keyword(s):

Sufficient Conditions ◽

Positive Equilibrium ◽

Auxiliary Result ◽

Piecewise Constant ◽

Trivial Equilibrium ◽

Stability Examination ◽

The Stability ◽

The authors consider a nonlinear epidemic equation by modeling it with generalized piecewise constant argument (GPCA). The authors investigate invariance region for the considered model. Sufficient conditions guaranteeing the existence and uniqueness of the solutions of the model are given by creating integral equations. An important auxiliary result giving a relation between the values of the unknown function solutions at the deviation argument and at any time t is indicated. By using Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), the stability of the trivial equilibrium is investigated in addition to the stability examination of the positive equilibrium transformed into the trivial equilibrium. Then sufficient conditions for the uniform stability and the uniform asymptotic stability of trivial equilibrium and the positive equilibrium are given.

Stability and oscillation for linear and nonlinear scalar equations with piecewise constant argument

Applicable Analysis ◽

10.1080/00036819208840071 ◽

1992 ◽

Vol 44 (1-2) ◽

pp. 113-125 ◽

Cited By ~ 6

Author(s):

Tetsuo Furumochi

Keyword(s):

Piecewise Constant ◽

Linear And Nonlinear ◽

Scalar Equations ◽

ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972713001020 ◽

2013 ◽

Vol 90 (1) ◽

pp. 99-112 ◽

Cited By ~ 1

Author(s):

LI-LI ZHANG ◽

HONG-XU LI

Keyword(s):

Differential Equations ◽

Existence And Uniqueness Theorem ◽

Piecewise Constant ◽

Exponential Dichotomies ◽

Almost Automorphic ◽

Greatest Integer Function ◽

Image Position ◽

AbstractUsing the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form $$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$ where $\lfloor \cdot \rfloor $ denotes the greatest integer function, and $A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $, $f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.

Emerging Applications of Differential Equations and Game Theory - Advances in Computer and Electrical Engineering ◽

The Stability of an Epidemic Model With Piecewise Constant Argument by Lyapunov-Razumikhin Method

10.4018/978-1-7998-0134-4.ch011 ◽

2020 ◽

pp. 236-259

Author(s):

Nur Cengiz ◽

Duygu Aruğaslan-Çinçin

Keyword(s):

Epidemic Model ◽

Sufficient Conditions ◽

Positive Equilibrium ◽

Piecewise Constant ◽

Trivial Equilibrium ◽

The Stability ◽

Theoretical Results ◽

The authors propose a nonlinear epidemic model by developing it with generalized piecewise constant argument (GPCA) introduced by Akhmet. The authors investigate invariance region for the considered model. For the taken model into consideration, they obtain a useful inequality concerning relation between the values of the solutions at the deviation argument and at any time for the epidemic model. The authors reach sufficient conditions for the existence and uniqueness of the solutions. Then, based on Lyapunov-Razumikhin method developed by Akhmet and Aruğaslan for the differential equations with generalized piecewise constant argument (EPCAG), sufficient conditions for the stability of the trivial equilibrium and the positive equilibrium are investigated. Thus, the theoretical results concerning the uniform stability of the equilibriums are given.

Behavior of the solutions of a partial differential equation with a piecewise constant argument

Filomat ◽

10.2298/fil1719931b ◽

2017 ◽

Vol 31 (19) ◽

pp. 5931-5943 ◽

Cited By ~ 3

Author(s):

Huseyin Bereketoglu ◽

Mehtap Lafci

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Partial Differential ◽

Piecewise Constant ◽

Oscillation Instability ◽

In this paper, we consider a partial differential equation with a piecewise constant argument. We study existence and uniqueness of the solutions of this equation. We also investigate oscillation, instability and stability of the solutions.

Asymptotically almost automorphic solutions of differential equations with piecewise constant argument

Open Mathematics ◽

10.1515/math-2017-0051 ◽

2017 ◽

Vol 15 (1) ◽

pp. 595-610

Author(s):

Hui-Sheng Ding ◽

Shun-Mei Wan

Keyword(s):

Differential Equations ◽

Uniqueness Theorem ◽

Systematic Study ◽

Difference System ◽

Existence And Uniqueness Theorem ◽

Piecewise Constant ◽

Almost Automorphic ◽

Abstract This paper is concerned with the existence and uniqueness of asymptotically almost automorphic solutions to differential equations with piecewise constant argument. To study that, we first introduce several notions about asymptotically almost automorphic type functions and obtain some properties of such functions. Then, on the basis of a systematic study on the associated difference system, the existence and uniqueness theorem is established. Compared with some earlier results, we do not assume directly that the Green’s function is a Bi-almost automorphic type function.